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I'm stuck in a proof: given that $E\subset \mathbb{R}^n$ is convex, prove that it is connected. I know that this can be proved easier using the concept of path-connectedness, but that's not I'm "allowed" to do. I need to use another definition: $E$ is connected if and only if it cannot be separatedby a pair of two relatively open sets.

My attempt:

Pick any $x, y\in E$. Since $E$ is convex, $tx+(1-t)y \subset E$ for $t\in [0,1]$. Suppose that $E$ is not connected. Then there exist non-empty sets $U$ and $V$, such that $U\cap V =\emptyset$, $U\cup V=E$, and $U$ and $V$ are relatively open in $E$, which implies that there exist open sets $A$ and $B$ such that $U=A\cap E$ and $V=B\cap E$.

Let $x \in U$ and $y \in V$.

(i) $U$ and $V$ are clearly not empty.

(ii) $U\cup V$...

Intuitively I understand that what needs to be shown is that $U \cup V \ne E$, but I'm stuck in trying to show it. Would appreciate some help.

sequence
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1 Answers1

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Hint: (I guess you are "allowed" to do this though it is really just the concept of path connectedness) Supposing for contradiction the existence of such $U$ and $V$ that disconnect $E$, and $x \in U$ and $y \in V$. Then the $U$ and $V$ also disconnect the line between $x$ and $y$. (Now use that a line is a continuous function from $[0,1]$, and that preimage commutes with set theoretic operations.)

Elle Najt
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  • I'm going to steal your "about me"! Describes me in the best possible way! In order to stick to the subject at hand XD, you should say something more rigorous about the last statement imo! – Maffred Jan 28 '16 at 01:16
  • AreaMan: Yes, I've though about it. That is, for some $t\in [0,1]$, the point on the line is not in $U$ and not in $V$, implying also that $U\cup V \ne E$. But the question is how to show it. – sequence Jan 28 '16 at 01:20
  • @sequence You should notice that the line is really a continuous function from $[0,1]$ into your space. Now continuous functions pull back opens, and preimage plays nicely with the kind of operations you want. – Elle Najt Jan 28 '16 at 01:21
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    @Maffred You actually stole it! I am truly flattered. :) – Elle Najt Jan 28 '16 at 01:22
  • @Maffred Of course, I do expect some form of compensation in the form of, say, an upvote on this answer. – Elle Najt Jan 28 '16 at 01:24
  • Well, It seems like the one over you wrote something more illuminating, even tho imo your solution contains his solution strictly! I vote for you! xD – Maffred Jan 28 '16 at 01:26